#8425337

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4185 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,5)(3,9,4,8)(6,10,14,13)(11,16,12,15)(17,18,22,25)(19,24,26,23)(20,29,30,21)(27,28,32,31)(33,34,40,44)(35,45,47,41)(36,43,38,42)(37,46,48,39);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(19,26)(23,24)(27,32)(28,31)(33,35,40,47)(34,41,44,45)(36,46,38,39)(37,42,48,43);(17,19)(18,23)(20,27)(21,28)(22,26)(24,25)(29,31)(30,32)(33,36,40,38)(34,42,44,43)(35,46,47,39)(37,41,48,45);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,19)(18,24)(20,27)(21,31)(22,26)(23,25)(28,29)(30,32)(33,37,40,48)(34,39,44,46)(35,43,47,42)(36,41,38,45);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,20)(18,21)(19,27)(22,30)(23,28)(24,31)(25,29)(26,32)(33,38,40,36)(34,43,44,42)(35,39,47,46)(37,45,48,41);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,21)(18,20)(19,28)(22,29)(23,27)(24,32)(25,30)(26,31)(33,39)(34,37)(35,36)(38,47)(40,46)(41,42)(43,45)(44,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332533);